coordination failures and the lender of last resort: was ... · discuss the interaction between...
TRANSCRIPT
Coordination Failures and the Lender of Last Resort:
Was Bagehot Right After All?
Jean-Charles RochetUniversite de Toulouse
Institut d’Economie Industrielleand
Xavier VivesINSEAD and ICREA-UPF
July 6, 2004
Abstract
The classical doctrine of the Lender of Last Resort, elaborated by Bagehot (1873), asserts thatthe central bank should lend to “illiquid but solvent” banks under certain conditions. Severalauthors have argued that this view is now obsolete: when interbank markets are efficient, asolvent bank cannot be illiquid. This paper provides a possible theoretical foundation for rescuingBagehot’s view. Our theory does not rely on the multiplicity of equilibria that arises in classicalmodels of bank runs. We build a model of banks’ liquidity crises that possesses a unique Bayesianequilibrium. In this equilibrium, there is a positive probability that a solvent bank cannot findliquidity assistance in the market. We derive policy implications about banking regulation(solvency and liquidity ratios) and interventions of the Lender of Last Resort. Furthermore, wefind that public (bail-out) and private (bail-in) involvement are complementary in implementingthe incentive efficient solution and that Bagehot’s Lender of Last Resort facility has to worktogether with institutions providing prompt corrective action and orderly failure resolution.Finally, we derive similar implications for an international Lender of Last Resort.
Keywords: Central bank policy, interbank market, prudential regulation, liquidity ratio, sol-vency ratio, transparency, prompt corrective action, orderly failure resolution, global games,supermodular games.
(JEL: G21, G28)
Acknowledgements: We are grateful to many colleagues and seminar participants at Bank of Italy, ESEMat Venice, ECB, Institute for Advanced Studies at Princeton, IMF, INSEAD, New York Fed, SverigesRiskbank, and UCL for helpful discussions and comments. Vives is grateful for support to the Pricewa-terhouse Coopers Initiative at INSEAD.
E-mail addresses: Rochet: [email protected]; Vives: [email protected]
1
1 Introduction
There have been several recent controversies about the need for a Lender of Last Resort (LLR)
both within national banking systems (central bank) and at an international level (IMF).1 The
concept of a LLR was elaborated in the XIXth century by Thornton (1802) and Bagehot (1873).
An essential point of the “classical” doctrine associated to Bagehot asserts that the LLR role is
to lend to “solvent but illiquid” banks under certain conditions.2
Banking crises have been recurrent in most financial systems. The LLR facility and deposit
insurance were instituted precisely to provide stability to the banking system and avoid the
consequences for the real sector. Indeed, financial distress may cause important damage to
the economy as the example of the Great Depression makes clear.3 Traditional banking panics
were eliminated with the LLR facility and deposit insurance by the end of the XIX century in
Europe, after the crisis of the 1930s in the US and also mostly in emerging economies, which
have suffered numerous crises until today.4 Modern liquidity crises associated to securitized
money or capital markets have also required the intervention of the LLR. Indeed, the Federal
Reserve intervened in the crises provoked by the failure of Penn Central in the US commercial
paper market in 1970, by the stock market crash of October 1987 and by Russia’s default in
1997 and subsequent collapse of LTCM (in the latter case a ”lifeboat” was arranged by the New
York Fed). For example, in October 1987 the Federal Reserve supplied liquidity to banks with
the discount window.5
1See for instance Calomiris (1998a,b), Kaufman (1991), Fischer (1999), Mishkin (1998), and Goodhartand Huang (1999a,b).
2The LLR should lend freely against good collateral, valued at pre-crisis levels, and at a penalty rate.Bagehot (1873), also presented for instance in Humphrey (1975) and Freixas et al. (1999).
3See Bernanke (1983) and Bernanke and Gertler (1989).4See Gorton (1988) for US evidence and Lindgren et al (1996) for evidence on other IMF member
countries.5See Folkerts-Landau and Garber (1992). See also Freixas et al. (2003) for a modeling of the interac-
tions between the discount window and the interbank market.
2
The function of the LLR of providing emergency liquidity assistance has been criticized for pro-
voking moral hazard on the banks’ side. Perhaps more importantly, Goodfriend and King (1988)
(see also Bordo (1990), Kaufman (1991) and Schwartz (1992)) remark that Bagehot’s doctrine
was elaborated at a time where financial markets were underdeveloped. They argue that, while
central banks interventions on aggregate liquidity (monetary policy) are still warranted, indi-
vidual interventions (banking policy) are not anymore: “with sophisticated interbank markets,
banking policy has become redundant”.Open market operations can provide sufficient liquidity
which is then allocated by the interbank market. The discount window is not needed. In other
words, Goodfriend and King argue that when financial markets are well-functioning, a solvent
institution cannot be illiquid. Banks can finance their assets with interbank funds, negotiable
certificates of deposit (CDs) and repurchase agreements (repos). Well informed participants in
this interbank market will make out liquidity from solvency problems. This view has conse-
quences also for the debate about the need of an international LLR. Indeed, Chari and Kehoe
(1998) claim, for example, that such an international LLR is not needed because the joint action
of the Federal Reserve, the European Central Bank and the Bank of Japan can take care of any
international liquidity problem.6
Those developments have led qualified observers to dismiss bank panics as a phenomenon of the
past and express confidence on the efficiency of financial markets, in particular the interbank
market, to resolve liquidity problems of financial intermediaries. This is based on the view that
participants in the interbank market are the best informed agents to ascertain the solvency of
an institution with liquidity problems.7
6Jeanne and Wyplosz (2001) compare the required size of an international LLR under the ”openmarket-monetary policy” and the ”discount window-banking policy” views.
7For example, Tommaso Padoa-Schioppa, member of the Executive Committee of the European Cen-tral Bank in charge of banking supervision, has gone as far as saying that classical bank runs may occuronly in textbooks, precisely because measures like deposit insurance and capital adequacy requirementshave been put in place. Furthermore, despite recognizing that ”rapid outflows of uninsured interbankliabilities” are less unlikely, Padoa-Schioppa states that ”However, since interbank counterparties are
3
The main objective of this article is to provide a theoretical foundation for Bagehot’s doctrine
in a model that fits the modern context of sophisticated and presumably efficient financial
markets. We are thinking of a short time horizon that corresponds to liquidity crises. We
shift emphasis from maturity transformation and liquidity insurance of small depositors to the
“modern” form of bank runs where large well-informed investors refuse to renew their credit
(CDs for example) on the interbank market. The decision not to renew credit may arise as a
result on an event (failure of Penn Central, October 1987 crash or LTCM failure) which puts
in doubt the repayment capacity of an intermediary or a number of intermediaries. The central
bank may then decide to provide liquidity to those troubled institutions. The question arises
about whether such intervention is warranted. At the same time it is debated whether central
banks should disclose the information they have on potential crisis situations (or the predictions
of their internal forecasting models) and what degree of transparency should a central bank’s
announcements have.8 We also hope to shed some light on these issues of transparency and
optimal disclosure of information by the central bank.
Since Diamond and Dybvig (1983) (and Bryant (1980)), banking theory has insisted on the
fragility of banks due to possible coordination failures between depositors (bank runs). However
it is hard to base any policy recommendation on their model, since it systematically possesses
multiple equilibria. Furthermore, a run equilibrium needs to be justified with the presence of
sunspots that coordinate the behavior of investors. Indeed, otherwise no one would deposit in a
bank that will be subject to run. This view of banking instability has been disputed by Gorton
(1985) and others who argue that crises are related to fundamentals and not to self-fulfilling
much better informed than depositors, this event would typically require the market to have a strong sus-picion that the bank is actually insolvent. If such a suspicion were to be unfounded and not generalised,the width and depth of today’s interbank market is such that other institutions would probably replace(possibly with the encouragement of the public authorities as described above) those which withdrawtheir funds” (Padoa-Schioppa (1999)).
8See, for example, Tarkka and Mayes (2000).
4
panics. In this view, crises are triggered by bad news about the returns to be obtained by the
bank. Gorton (1988) studies panics in the National Banking Era in the US and concludes that
crises were predictable by indicators of the business cycle.9 There is an ongoing empirical debate
about whether crises are predictable and their relation to fundamentals.10
Our approach is inspired by Postlewaite and Vives (1987), who display an incomplete information
model with a unique Bayesian equilibrium with a positive probability of bank runs11, and the
model is adapted from the ”global game” analysis of Carlsson and Van Damme (1993) and Morris
and Shin (1998).12 This approach builds a bridge between the ”panic” and ”fundamentals”
view of crises by linking the probability of occurrence of a crisis to the fundamentals. A crucial
property of the model is that, when the private information of investors is precise enough, the
game among them has a unique equilibrium. Moreover, at this unique equilibrium there is an
intermediate interval of values of the bank’s assets for which, in the absence of intervention by the
central bank, the bank is solvent but can fail by the fact that a too large proportion of investors
withdraw their money. In other words, in this intermediate range for the fundamentals there is
the potential for a coordination failure. Furthermore, the range in which such a coordination
failure occurs diminishes with the ex ante strength of fundamentals.
Given that this equilibrium is unique and based on the fundamentals of the bank, we are able
to provide some policy recommendations on how to avoid such failures. More specifically, we
discuss the interaction between ex-ante regulation of solvency and liquidity ratios and ex-post9The phenomenon has been theorized in the literature on information-based bank runs such as Chari
and Jagannathan (1988), Jacklin and Bhattacharya (1988) and Allen and Gale (1998).10See also Kaminsky et al (1999) and Radelet and Sachs (1998) for perspectives on international crises.11However, the model of Postlewaite and Vives (1987) differs from our model here in several respects.
In particular, in Postlewaite and Vives there is no uncertainty about the fundamental value of thebanks’assets (no solvency problems) but incomplete information about the liquidity shocks suffered bydepositors. The uniqueness of equilibrium in their case comes from a more complex specification oftechnology and liquidity shocks for depositors than in Diamond and Dybvig (1983).
12See also Heinemann and Illing (2000) and Corsetti et al (2000).
5
provision of emergency liquidity assistance. It is found that liquidity and solvency regulation can
solve the coordination problem but typically the cost is too high in terms of foregone returns.
This means that prudential measures have to be complemented with emergency discount window
loans.
We endogenize banks’ short-term debt structure as a way to discipline bank managers because
of a moral hazard problem. The framework allows us to discuss early closure policies of banks
and the interaction of the LLR, prompt corrective action and orderly resolution of failures.
We can study then the adequacy of Bagehot’s doctrine in a richer environment and derive the
complementarity between public (LLR and other facilities) and private (market) involvement in
crisis resolution.
Finally, we provide a reinterpreation of the model in terms of the banking sector of a small open
economy and derive lessons for a international LLR facility.
The rest of the article is organized as follows:
• Section 2 presents the model.
• Section 3 discusses runs and solvency.
• Section 4 characterizes the equilibrium of the game between investors.
• Section 5 studies the properties of this equilibrium and the effect of prudential regulation
on coordination failure.
• Section 6 makes a first pass at the LLR policy implications of our model and the relations
with Bagehot’s doctrine.
• Section 7 shows how to endogenize the liability structure and proposes a welfare-based
LLR facility with attention to crisis resolution.
6
• Section 8 provides the international reinterpretation of the model and discusses the role
of an international LLR and associated facilities.
• Concluding remarks end the paper.
2 The Model
Consider a market with three dates: τ = 0, 1, 2. At date τ = 0 the bank possesses own funds E,
and collects uninsured wholesale deposits (CDs for example) for some amount D0, normalized
to 1. These funds are used in part to finance some investment I in risky assets (loans), the rest
being held in cash reserves M . Under normal circumstances, the returns RI on these assets are
collected at date τ = 2, the CDs are repaid, and the stockholders of the bank get the difference
(when it is positive). However, early withdrawals may occur at an interim date τ = 1, following
the observation of private signals on the future realization of R. If the proportion x of these
withdrawals exceeds the cash reserves M of the bank, the bank is forced to sell some of its assets.
To summarize our notation, the bank’s balance sheet at τ = 0 is represented as follows:
I D0 = 1M E
where:
• D0 (= 1) is the volume of uninsured wholesale deposits, normally repaid at τ = 2 but
that can also be withdrawn at τ = 1. The nominal value of deposits upon withdrawal is
D ≥ 1 independently of the withdrawal date. So, early withdrawal entails no cost for the
depositors themselves (when the bank is not liquidated prematurely).
• E represents the value of equity (or more generally long term debt; it may also include
7
insured deposits13).
• I denotes the volume of investment in risky assets, which have a random return R at
τ = 2.
• Finally, M is the amount of cash reserves (money) held by the bank.
We assume that the withdrawal decision is delegated to fund managers who typically prefer to
renew the deposits (i.e. not to withdraw early) but are penalized by the investors if the bank
fails. Suppose that fund managers obtain a benefit B > 0 if they get the money back or if they
withdraw and the bank fails. They get nothing otherwise. However, to withdraw involves a cost
C > 0 for the managers (for example because their reputation suffers if they have to recognize
that they have made a bad investment). The net expected benefit of withdrawing is B −C > 0
while the one of not withdrawing is (1 − P )B, where P is the probability that the bank fails.
Accordingly, fund managers adopt the following behavioral rule: withdraw if and only if they
anticipate P > γ = C/B, where γ ∈ (0, 1).14
At τ = 1, fund manager i privately observes a signal si = R+εi, where the εis are i.i.d. and also
independent of R. As a result, a proportion x of them decides to “withdraw” (i.e. not to renew
their CDs). By assumption there is no other source of financing for the bank (except maybe the
central bank, see below) so if x > MD , the bank is forced to sell a volume y of assets:15 if the
needed volume of sales y is greater than the total of available assets I the bank fails at τ = 1.13If they are fully insured, these deposits have no reason to be withdrawn early and can thus be
assimilated to stable resources.14The fact that fund managers make the withdrawal decisions is realistic in the interbank market, as
well as in the international interpretation in Section 8. Alternatively, we could model the decisions ofinvestors directly at the cost of further assumptions and complicating the analysis with no further benefitfor our purposes. See Goldstein and Pauzner (2000) for an analysis of runs with depositors investingdirectly.
15These sales are typically accompanied with a repurchase agreement or repo. They are thus equivalentto a collateralized loan.
8
If not, the bank continues until date 2. Failure occurs at τ = 2 whenever
R(I − y) < (1 − x)D. (1)
Our modeling tries to capture in the simplest possible way the main institutional features of
modern interbank markets. In our model, banks essentially finance themselves by two com-
plementary sources: stable resources (equity and long term debt) and uninsured short term
deposits (or CDs), which are uncollateralized and involve fixed repayments. However, in case
of a liquidity shortage at date 1, banks also have the possibility to sell some of their assets
(or equivalently borrow against collateral) on the repo market. This secondary market for bank
assets is assumed to be informationally efficient, in the sense that the secondary price aggregates
the decentralized information of investors about the quality of the bank’s assets.16 Therefore we
assume that the resale value of the bank’s assets depends on R. However banks cannot obtain
the full value of these assets but only a fraction of this value 11+λ , with λ > 0. Accordingly the
volume of sales needed to face withdrawals x is given by:
y = (1 + λ)[xD − M ]+
R
where (xD − M)+ = max(0, xD − M).
The parameter λ measures the cost of ”fire sales” in the secondary market for bank assets. It is
crucial for our analysis, and can be explained by considerations of asymmetric information or
liquidity problems.17
Indeed, asymmetric information problems may translate into limited commitment of future
cash flows (as in Hart and Moore (1994) or Diamond and Rajan (2001)), moral hazard (as in16We can imagine for instance that the bank organizes an auction for the sale of its assets. If there is a
large number of bidders and their signals are (conditionally) independent, the equilibrium price p of thisauction will be a deterministic function of R .
17For a similar assumption in a model of an international lender of last resort, see Goodhart and Huang(1999b).
9
Holmstrom and Tirole (1997)), or adverse selection (as in Flannery (1996)). We have chosen
to stress the last explanation, because it gives a simple justification for the superiority of the
central bank over financial markets in the provision of liquidity to banks in trouble. The presence
of an adverse selection discount in credit markets is well established (see, e.g, Broecker (1990)
and Riordan (1993)). Flannery (1996) presents a specific mechanism which explains why the
secondary market for banks assets may be plagued by a winner’s curse which induces a fire
sales premium. He argues, furthermore, that this fire sales premium is likely to be higher during
crises, given that investors are then probably more uncertain about the precision of their signals.
This makes the winner’s curse more severe because it is more difficult to identify good from bad
risks. The superiority of the central bank resides in its large financial capacity, and thus its
ability to eliminate the adverse selection problem by buying the entire portfolio at a unit price
of R.
The parameter λ can also be interpreted as a liquidity premium, i.e. the interest margin that
the market requires for lending on a short notice.18 In a generalized banking crisis we would
have a liquidity shortage implying a large λ. Interpreting λ as a market rate, λ can also spike
temporarily in response to exogenous events, such as September 11.
In our model we will be thinking mostly of the financial distress of an individual bank (a bank
is close to insolvency when R is small) although for correlated enough portfolio returns of the
banks the interpretation could be broadened (see also the interpretation in an international
context in Section 8).
Operations on interbank markets do not involve any physical liquidation of bank assets. However,
we will show that when a bank is close to insolvency (R small) or when there is a liquidity18See Allen and Gale (1998) for a model where costly asset sales arise due to the presence of liquidity
constrained speculators in the resale market.
10
shortage (λ large) the interbank markets do not suffice to prevent early closure of the bank. Early
closure involves the physical liquidation of assets and this is costly. We model this liquidation
cost (not to be confused with the fire sales premium λ) as proportional to the future returns on
the bank’s portfolio. If the bank is closed at τ = 1, the (per unit) liquidation value of its assets
is νR, with ν � 11+λ .
3 Runs and solvency
We focus in this section on some features of banks’ liquidity crises that cannot be properly taken
into account within the classical Bryant-Diamond-Dybvig (BDD) framework. In doing so we
take the banks’ liability structure (and in particular the fact that an important fraction of these
liabilities can be withdrawn on demand) as exogenous. A possible way to endogenize the bank’s
liability structure is to introduce a disciplining role for liquid deposits. In Section 8 we explore
such an extension.
We adopt explicitly the short time horizon (say 2 days) that corresponds to liquidity crises. This
means that we shift the emphasis from maturity transformation and liquidity insurance of small
depositors to the “modern” form of bank runs, i.e. large investors refusing to renew their CDs
on the interbank market.
A second element that differentiates our model from BDD is that our bank is not a mutual bank,
but a corporation that acts in the best interest of its stockholders. This allows us to discuss the
role of equity and the articulation between solvency requirements and provision of emergency
liquidity assistance. In Section 7 we endogenize the choice of assets by the bank through the
monitoring effort of banks’managers (first order stochastic dominance), but we take as given
the amount of equity E. It would be interesting to extend our model, and endogenize the level
of equity, in order to capture the impact of leverage on the riskiness of assets chosen by banks
11
(second order stochastic dominance). In this model however, both the amount of equity and the
riskiness of assets are taken as given.
As a consequence of our assumptions, the relation between x, the proportion of early with-
drawals, and the failure of the bank is different from that in BDD. To see this, let us recapitulate
the different cases:
• xD ≤ M : there are no sale of assets at τ = 1. In this case there is failure at τ = 2 if and
only if
RI + M < D ⇔ R < Rs =D − M
I= 1 − 1 + E − D
I.
Rs can be interpreted as the solvency threshold of the bank. Indeed, if there are no
withdrawals at τ = 1 (x = 0), the bank fails at τ = 2 if and only if R < Rs. The threshold
Rs is a decreasing function of the solvency ratio EI .
• M < xD ≤ M + RI1+λ : there is a partial sale of assets at τ = 1. Failure occurs at τ = 2 if
and only if
RI − (1 + λ)(xD − M) < (1 − x)D ⇔ R < Rs + λxD − M
I= Rs
[1 + λ
xD − M
D − M
].
This formula illustrates how, because of the premium λ , solvent banks can fail when the
proportion x of early withdrawals is too big19. Notice however an important difference with
BDD: when the bank is ”supersolvent” (R > (1 + λ)Rs) it can never fail, even if everybody
withdraws (x = 1).
• Finally, when xD > M + RI1+λ , the bank is closed at τ = 1 (early closure).
19Note that we can interpret that to obtain resources xD − M > 0 we need to liquidate a fraction ofthe portfolio µ = xD−M
RI (1 + λ) and therefore at τ = 2 we have left R(1 − µ)I = RI − (1 + λ)(xD − M).
12
The failure thresholds are summarized in Figure 1 below:
� R
failure dependson x
Rs (1 + λ)Rsalwaysfailure
no failure (even ifeverybody withdraws)
Figure 1
Several comments are in order:
• In our model, early closure is never ex post efficient because to physically liquidate assets
is costly. However, as discussed in Section 8, early closure may be ex ante efficient to
discipline bank managers and induce them to exert effort.
• The perfect information benchmark of our model (where R is common knowledge at
τ = 1) has different properties than in BDD: the multiplicity of equilibria only arises in
the median range Rs ≤ R ≤ (1 + λ)Rs. When R < Rs everybody runs (x = 1), when
R > (1 + λ)Rs nobody runs (x = 0) and only in the intermediate region both equilibria
coexist.20 This pattern is crucial for being able to select a unique equilibrium through the
introduction of private noisy signals (when noise is not too important, as in Morris and
Shin (1998)).21
20When R < Rs fund managers get B − C > 0 by withdrawing and nothing by waiting. WhenR > (1+λ)Rs fund managers by withdrawing get B−C and by waiting B. Note that if depositors madedirectly the investment decisions the equilibria would be the same provided that there is a small cost ofwithdrawal.
21Goldstein and Pauzner (2000) adapt the same methodology to the BDD model, in which the perfectinformation game always has two equilibria, even for very large R. Accordingly, they have to make anextra assumption, namely that ”there exists an external lender who would be willing to buy any amountof the investment... if she knew for sure that the long-run return was excessively high” (Goldstein andPauzner (2000), p.11), in order to obtain a unique equilibrium in the presence of private signals withsmall noise. See also Morris and Shin (2000).
13
The different regimes of the bank, as a function of R and x, are represented in Figure 2.
�
�
R
��
��
��
��
��
��
��
��
��
��
��
��
���
��
��
��
��
��
��
��
��
x
1
M/D
Rs (1 + λ)Rs
No failure
Completeliquidation at τ = 1
no liquidation at τ = 1
Failure at τ = 2
Failure at τ = 2at τ = 1
liquidationPartial
Figure 2
The critical value of R below which the bank is closed early is given by:
Rec(x) = (1 + λ)(xD − M)+
I.
The critical value of R below which the bank fails is given by:
Rf (x) = Rs + λ(xD − M)+
I. (2)
The parameters Rs, M and I are not independent. Since we want to study the impact of
prudential regulation on the need for central bank intervention, we will focus on Rs (a decreasing
function of the solvency ratio E/I ) and m = MD (the liquidity ratio). Replacing I by its value
D−MRs
, we obtain:
14
Rec(x) = Rs(1 + λ)(x − m)+
1 − m, and
Rf (x) = Rs(1 + λ(x − m)+
1 − m).
It should be obvious that Rec(x) < Rf (x) since early closure implies failure while the converse
is not true (see Figure 2).
4 Equilibrium of the investors’ game
In order to simplify the presentation we concentrate on “threshold” strategies, in which each
fund manager decides to withdraw if and only if his signal is below some threshold t.22 As we
will see later this is without loss of generality. For a given R, a fund manager withdraws with
probability
Pr[R + ε < t] = G(t − R),
where G is the c.d.f. of the random variable ε. Given our assumptions, this probability also
equals the proportion of withdrawals x(R, t).
A fund manager withdraws if and only if the probability of failure of the bank (conditional
on the signal s received by the manager and the threshold t used by other managers) is large
enough. That is, P (s, t) > γ , where
P (s, t) = Pr[failure|s, t]
= Pr[R < Rf (x(R, t))|s].22It is assumed that the decision on whether to witdraw is taken before the secondary market is
organized and thus before fund managers have the opportunity to learn about R from the secondaryprice. (On this issue see Atkeson’s comments on Morris and Shin (2000).)
15
Before we analyze the equilibrium of the investor’s game let us look at the region of the plane
(t, R) where failure occurs. For this, transform Figure 2 by replacing x by x(R, t) = G(t − R).
We obtain Figure 3 below.
� t
�
R
Rs(1 + λ)
Rs
t0
R = RF (t)
Failure caused
by insolvency
Failure caused
by illiquidity
Figure 3
Notice that RF (t), the critical R that triggers failure is equal to the solvency threshold Rs when
t is low and fund managers are confident about the strength of fundamentals:
RF (t) = Rs if t ≤ t0 = Rs + G−1(m).
However, for t > t0, RF (t) is an increasing function of t and is defined implicitly by
R = Rs(1 + λ[G(t − R) − m
1 − m]).
Let us denote by G(.|s) the c.d.f. of R conditional on signal s :
G(r|s) = Pr[R < r|s].
16
Then given the definition of RF (t)
P (s, t) = Pr[R < RF (t)|s] = G(RF (t)|s) (3)
It is natural to assume that G(r|s) is decreasing in s: the higher s, the lower the probability
that R lies below any given threshold r. Then it is immediate that P is decreasing in s and
nondecreasing in t: ∂P∂s < 0 and ∂P
∂t ≥ 0. This means that the depositors’ game is one of strategic
complementarities. Indeed, given that other fund managers use the strategy with threshold t
the best response of a manager is to use a strategy with threshold s : withdraw if and only if
P (s, t) > γ or equivalently if and only if s < s where P (s, t) = γ. Let s = S(t). Now we have
that S′ = − ∂P/∂t∂P/∂s ≥ 0 : a higher threshold t by others induces a manager to use also a higher
threshold.
The strategic complementarity property holds for general strategies. For a fund manager all
that matters is the conditional probability of failure for a given signal and this depends only on
aggregate withdrawals. Recall that the differential payoff to a fund manager for withdrawing
over not withdrawing is given by PB − C where C/B = γ. A strategy for a fund manager is
a function a(s) ∈ {not withdraw, withdraw} . If more managers withdraw then the probability
of failure conditional on receiving signal s increases. This just means that the payoff to a fund
manager displays increasing differences with respect to the actions of others. The depositor’s
game is a supermodular game and there will exist a largest and a smallest equilibrium. In fact,
the game is symmetric (that is, exchangeable against permutations of the players) and there-
fore the largest and smallest equilibria are symmetric.23 At the largest equilibrium every fund
manager withdraws in the largest number of occasions, at the smallest equilibrium every fund
manager withdraws in the smallest number of occasions. The largest (smallest) equilibrium can23See Remark 15, p.34 in Vives (1999). See also Chapter 2 in the same reference for an exposition of
the theory of supermodular games.
17
be identified then with the highest (lowest) threshold strategy t(t).24 These extremal equilibria
bound the set of rationalizable outcomes. That is, strategies outside this set can be eliminated
by iterated deletion of dominated strategies.25 We will make assumptions so that t = t and
equilibrium will be unique.
The threshold t = t∗ corresponds to a (symmetric) Bayesian Nash equilibrium if and only if
P (t∗, t∗) = γ. Indeed, suppose that funds managers use the threshold strategy t∗. Then for
s = t∗, P = γ and since P is decreasing in s for s < t∗ we have that P (s, t∗) > γ and the
manager withdraws. Conversely, if t∗ is a (symmetric) equilibrium then for s = t∗ there is
no withdrawal and therefore P (t∗, t∗) ≤ γ. If P (t∗, t∗) < γ then by continuity for s close but
less than t∗ we would have P (s, t∗) < γ , a contradiction. It is clear then that the largest
and the smallest solutions to P (t∗, t∗) = γ correspond respectively to the largest and smallest
equilibrium.
An equilibrium can also be characterized by a couple of equations in two unknowns (a withdrawal
threshold t∗and a failure threshold R∗):
G(R∗|t∗) = γ, and (4)
R∗ = Rs(1 + λ[G(t∗ − R∗) − m
1 − m]+). (5)
Equation (4) states that conditionally on observing a signal s = t∗, the probability that R < R∗
is γ. Equation (5) states that, given a withdrawal threshold t∗, R∗ is the critical return (i.e. the24The extremal equilibria can be found with the usual algorithm in a supermodular game (Vives
(1990)), starting at the extremal points of the strategy sets of players and iterating using the bestresponses. For example, to obtain t let all investors withdraw for any signal received (that is, start fromt0 = + ∞ and x = 1) and applying iteratively the best response S(·) of a player obtain a decreasingsequence tk that converges to t. Note that S(+ ∞) = t1 < + ∞ where t1 is the unique solution toP (t,+∞) = G(Rs(1 + λ)|t) = γ given that G is (strictly) decreasing in t. The extremal equilibria are instrategies monotone in type, which with two actions means that the strategies are of the threshold type.The game among mutual fund managers is an example of a monotone supermodular game for which,according to Van Zandt and Vives (2003), extremal equilibria are monotone in type.
25See Morris and Shin (2000) for an explicit demonstration of the outcome of iterative elimination ofdominated strategies in a similar model.
18
one below which failure occurs). Equation (5) implies that R∗ belongs to [Rs, (1+λ)Rs]. Notice
that early closure occurs whenever x(R, t∗)D > M + IR1+λ ,where x(R, t∗) = G(t∗ − R). This
happens if and only if R is smaller than some threshold REC(t∗). We will have that REC(t∗) <
R∗ since early closure implies failure, while the converse is not true, as remarked before.
In order to simplify the analysis of this system we are going to make distributional assumptions
on returns and signals. More specifically, we will assume that the distributions of R and ε are
normal, with respective means R and 0, and respective precisions (i.e. inverse variances) α and
β. Denoting by Φ the c.d.f. of a standard normal distribution the equilibrium is characterized
then by a pair ( t∗, R∗) such that:
Φ(√
α + βR∗ − αR + βt∗√α + β
)= γ, (6)
and
R∗ = Rs
(1 + λ
[Φ(
√β(t∗ − R∗)) − m
1 − m
]+
). (7)
We now can now state our first result.
Proposition 1 When β (the precision of the private signal of investors) is large enough relative
to α (prior precision), there is a unique t∗ such that P (t∗, t∗) = γ. The investor’s game has then a
unique (Bayesian) equilibrium. In this equilibrium, fund managers use a strategy with threshold
t∗.
Proof of Proposition 1: We show that ϕ(s) def= P (s, s) is decreasing for
β ≥ β0def= 1
2π
(λαD
I
)2with I = D−M
Rs. Under our assumptions R conditional on signal realization
s follows a normal distribution N(αR+βsα+β , 1
α+β ). Denoting by Φ the c.d.f. of a standard normal
19
distribution, it follows that
ϕ(s) = P (s, s) = Pr[R < RF (s)|s]
= Φ[√
α + βRF (s) − αR + βs√α + β
]. (8)
This function is clearly decreasing for s < t0 since, in this region, we have RF (s) ≡ Rs. Now if
s > t0, RF (s) is increasing and its inverse is
tF (R) = R +1√β
Φ−1
(I
λD(R − Rs) + m
).
The derivative of tF is
t′F (R) = 1 +1√β
I
λD
[Φ′
(Φ−1
(I
λD(R − Rs) + m
))]−1
.
Since Φ′ is bounded above by 1√2π
, t′F is bounded below:
t′F (R) ≥ 1 +√
2π
β
I
λ.
Thus
R′F (s) ≤
[1 +
√2π
β
I
λD
]−1
.
Given formula (8), ϕ(s) will be decreasing provided that
√α + β
(1 +
√2π
β
I
λD
)−1
≤ β√α + β
,
which, after simplification, gives: β ≥ 12π
(λαD
I
)2. If this condition is satisfied, there is at most
one equilibrium. Existence is easily shown. When s is small RF (s) = Rs and formula (8) implies
that lims→−∞ ϕ(s) = 1. On the other hand, when s → +∞, RF (s) → (1 + λ)Rs and ϕ(s) → 0.
20
The limit equilibrium when β tends to infinity can be characterized as follows: From equation
(6) we have that limβ→+∞√
β(R∗ − t∗) = Φ−1(γ). Given that Φ {−z} = 1 − Φ {z} we obtain
from formula (7) that in the limit t∗ = R∗ = Rs(1 + λ1−m [max {1 − γ − m, 0}]). The critical
cutoff R∗ is decreasing with γ and ranges from Rs for γ ≥ 1−m to (1+λ)Rs for γ = 0. It is also
nonincreasing in m. As we establish in the next section, these features of the limit equilibrium
are also valid for β ≥ β0.
It is worth noting also that with a diffuse prior (α = 0), the equilibrium is unique for any private
precision of investors (indeed, we have that β0 = 0). From (6) and (7) we obtain immediately
that R∗ = Rs(1 + λ1−m [max {1 − γ − m, 0}]) and t∗ = R∗ − Φ−1(γ)√
β. Both the cases β → +∞ and
α = 0 have in common that each investor faces the maximal uncertainty about the behavior of
other investors at the switching point si = t∗. Indeed, it can be easily checked that in either
case the distribution of the proportion x(R, t∗) = Φ(√
β(t∗ − R)) of investors withdrawing is
uniformly distributed over [0, 1] conditional on si = t∗. This contrasts with the certainty case
with multiple equilibria when R ∈ (Rs, (1 + λ)Rs) where, for example, in a run equilibrium an
investor thinks that with probability one all other investors will withdraw. It is precisely the
need to entertain a wider range of behavior of other investors in the incomplete information
game that pins down a unique equilibrium as in Carlsson and Van Damme (1993) or Postlewaite
and Vives (1987).
Public signals and transparency The analysis could be easily extended to allow for fund
managers to have access to a public signal v = R + η, where η ∼ N(0, 1
βp
)is independent from
R and from the error terms εi of the private signals. The only impact of the public signal is
to replace the unconditional moments R and 1α of R by its conditional moments taking into
account the public signal v. A disclosure of a signal of high enough precision will imply the
21
existence of multiple equilibria much in the same way as a precise enough prior would do.
The public signal could be provided by the central bank. Indeed, the central bank typically has
information about banks that the market does not have (and, conversely, market participants
have also information complementary to the central bank knowledge).26 The model allows for
the information structures of the central bank and investors to be non-nested. Our discussion
has then a bearing on the slippery issue of the optimal degree of transparency of central bank
announcements. Indeed, Alan Greenspan has become famous for his oblique way of saying
things, fostering an industry of ”Greenspanology” or interpretation of his statements. Our
model may rationalize oblique statements by central bankers that seem to add noise to a basic
message. Precisely because the central bank may be in a unique position to provide information
that becomes common knowledge, it has the capacity to destabilize expectations in the market
(which in our context means to move the interbank market to a regime of multiple equilibria).
By fudging the disclosure of information, the central bank makes sure that somewhat different
interpretations of the release will be made, preventing destabilization.27 Indeed, while in the
initial game without a public signal we may well be in the uniqueness region, adding a precise
enough public signal we will have three equilibria. At the interior equilibrium we have a similar
result than with no public information but run and no-run equilibria also exist. We may therefore
end up in an ”always run” situation when disclosing (or increasing the precision of) the public
signal while the economy was sitting in the interior equilibrium without public disclosure. In
other words, public disclosure of a precise enough signal may be destabilizing. This means that
a central bank that wants to avoid entering in the ”unstable” region may have to add noise to
its signal if the signal is ”too” precise.28
26See Peek et al (1999), De Young et al (1998), and Berger et al (1998).27The potential damaging effects of public information is a theme also developed in Morris and Shin
(2001).28See Hellwig (2002) for a treatment of the multiplicity issue.
22
5 Coordination failure and prudential regulation
For β large enough, we have just seen that there exists a unique equilibrium whereby investors
adopt a threshold t∗ characterized by
Φ(√
α + βRF (t∗) − αR + βt∗√α + β
)= γ,
or
RF (t∗) =1√
α + β
(Φ−1(γ) +
αR + βt∗√α + β
). (9)
For this equilibrium threshold, the failure of the bank will occur if and only if:
R < RF (t∗) = R∗.
This means that the bank fails if and only if fundamentals are weak, R < R∗. When R∗ > Rs we
have an intermediate interval of fundamentals R ∈ [Rs, R∗) where there is a coordination failure:
the bank is solvent but illiquid. The occurrence of a coordination failure can be controlled by
the level of the liquidity ratio m as the following proposition shows.
Proposition 2 There is a critical liquidity ratio of the bank m such that for m ≥ m we have
that R∗ = Rs, which means that only insolvent banks fail (there is no coordination failure).
Conversely, for m < m we have that R∗ > Rs. This means that for R ∈ [Rs, R∗) the bank is
solvent but illiquid (there is a coordination failure).
Proof of Proposition 2: For t∗ ≤ t0 = Rs + 1√βΦ−1(m), the equilibrium occurs for R∗ = Rs.
By replacing in formula (6) this gives:
(α + β)Rs ≤√
α + βΦ−1(γ) + αR + βRs +√
βΦ−1(m),
23
which is equivalent to:
Φ−1(m) ≥ α√β
(Rs − R) −√
1 +α
βΦ−1(γ). (10)
Therefore, the coordination failure disappears when m ≥ m, where
m = Φ(
α√β
(Rs − R) −√
1 +α
βΦ−1(γ)
).
Notice that, since Rs is a decreasing function of EI , the critical liquidity ratio m decreases when
the solvency ratio EI increases.29
The equilibrium threshold return R∗ is determined (when (10) is not satisfied) by the solution
to:
φ(R) = α(R − R) −√
βΦ−1
(1 − m
λRs(R − Rs) + m
)−
√α + βΦ−1(γ) = 0. (11)
When β ≥ β0, φ′(R) < 0 and the comparative statics properties of the equilibrium threshold R∗
are straightforward. Indeed, we have that ∂φ/∂m < 0, ∂φ/∂Rs > 0, ∂φ/∂λ > 0, ∂φ/∂γ < 0 and
∂φ/∂R < 0. The following proposition states the results.
Proposition 3 Comparative statics of R∗(and of the probability of failure):
• R∗ is a decreasing function of the liquidity ratio m and the solvency (E/I) of the bank, of
the critical withdrawal probability γ and of the expected return on the bank’s assets R.
• R∗ is an increasing function of the fire sales premium λ and of the face value of debt D.
29More generally, it is easy to see that in our model, the regulator can control the probabilities ofilliquidity (Pr(R < R∗)) and insolvency (Pr(R < Rs)) of the bank by imposing appropriate levels ofminimum liquidity and solvency ratios.
24
We have thus that stronger fundamentals, as indicated by a higher prior mean R also imply a
lower likelihood of failure. In contrast, a higher fire sales premium λ increases the incidence of
failure. Indeed, for a higher λ a larger portion of the portfolio must be liquidated to meet the
requirements of withdrawals. We have also that R∗ is decreasing with the critical withdrawal
probability γ and as γ → 0, R∗ → (1 + λ)Rs.
A similar analysis applies to changes in the precision of the prior α and of the private information
of investors β. Assume that γ = C/B < 1/2. Indeed, we should expect that the cost of
withdrawal C is small in relation to the continuation benefit for the fund managers B. If
γ < 1/2 it is easy to see that for large β
• and bad prior fundamentals (R low), increasing α increases R∗(more precise prior infor-
mation about a bad outcome worsens the coordination problem)30; and
• increasing β decreases R∗.31
6 Coordination failure and LLR policy
The main contribution of our paper so far has been to show the theoretical possibility of a
solvent bank being illiquid, due to a coordination failure on the interbank market. We are now30The effect of an increase in the precision of the prior α is potentially ambiguous. This is so because
∂φ/∂α = R∗ − R− Φ−1(γ)
2√
α+β, whose sign depends on whether R∗ � R and γ � 1/2 (recall that Φ−1(γ) � 0
as γ � 1/2). If γ < 1/2 and R∗ > R we have that ∂φ/∂α > 0. In consequence, increasing α will increaseR∗. It follows also that ∂ Pr[R < R∗]/∂α > 0. On the other hand, when the prior fundamentals are good(R high) and R∗ < R the outcome is ambiguous unless R∗ << R, in which case ∂φ/∂α < 0. Then a moreprecise prior information about a very good outcome alleviates the coordination problem. It follows alsothat ∂ Pr[R < R∗]/∂α < 0.
31The sign of {∂φ/∂β} depends on the sign of Φ−1(
1−mλRs
(R − Rs) + m)and of Φ−1(γ) and we may
have 1−mλRs
(R − Rs) + m � 1/2 and/or γ � 1/2. For example, for β large enough it can be seenthat sign {∂φ/∂β} = sign Φ−1(γ). For β large we have that, for R = R∗, sign {∂φ/∂β} = sign{
Φ−1(γ)2 ( 1√
β− 1√
α+β)}
= sign Φ−1(γ). Then an improved precision of private signals decreases (in-creases) R∗ and the failure rate, if the relative cost of withdrawal for the fund managers is small, γ < 1/2(large, γ > 1/2).
25
going to explore the lender of last resort policy of the central bank and present a scenario where
it is possible to give a theoretical justification to Bagehot’s doctrine.
We start by considering a simple central bank objective: Eliminate the coordination failure with
minimal involvement. The instruments at the disposal of the central bank are the liquidity ratio
m and intervention in the form of open market or discount window operations.32
We have shown in Section 5 that a high enough liquidity ratio m eliminates the coordination
failure altogether by inducing R∗ = Rs. This is so for m ≥ m. However, it is likely that imposing
m ≥ m might be too costly in terms of foregone returns (recall that I + M = 1 + E, where I
is the investment in the risky asset). In Section 7 we analyze a more elaborate welfare-oriented
objective and endogenize the choice of m. We look now at forms of central bank intervention
that can eliminate the coordination failure when m < m.
Let us see how central bank liquidity support can eliminate the coordination failure. Suppose
the central bank announces it will lend at rate r ∈ (0, λ), and without limits, but only to solvent
banks. The central bank is not allowed to subsidize banks and is assumed to observe R. The
knowledge of R may come from the supervisory knowledge of the central bank or perhaps by
observing the amount of withdrawals of the bank.33 Then the optimal strategy of a (solvent)
commercial bank will be to borrow exactly the liquidity it needs, i.e. D(x − m)+. Whenever
x − m > 0, failure will occur at date 2 if and only if:
RI
D< (1 − x) + (1 + r)(x − m).
32Open market operations typically involve performing a repo operation with primary security dealers.The Federal Reserve auctions a fixed amount of liquidity (reserves) and, in general, does not accept bidsby dealers below the Federal funds Rate target.
33The empirical evidence points at the superiority of the central bank information because of its accessto supervisory data (Peek et al. (1999), for example). Similarly, Romer and Romer (2000) find evidenceof a superiority of the Federal Reserve over commercial forecasters in forecasting inflation.
26
Given that DI = Rs
1−m , we obtain that failure at t = 2 will occur if and only if:
R < Rs(1 + r(x − m)+
1 − m).
This is exactly analogous to our previous formula giving the critical return of the bank, only that
the interest rate r replaces the liquidation premium λ. As a result, this type of intervention will
be fully effective (yielding R∗ = Rs) only when r is arbitrarily close to zero. It is worth to remark
that central bank help in the amount D(x−m)+ whenever the bank is solvent (R > Rs) and at a
very low rate avoids early closure, and the central bank loses no money because the loan can be
repaid at τ = 2. Note also that whenever the central bank lends at a very low rate the collateral
of the bank is evaluated under ”normal circumstances”, that is when there is no coordination
failure. Consider as an example the limit case of β tending to infinity. The equilibrium with no
central bank help is then t∗ = R∗ = Rs(1 + λ1−m [max {1 − γ − m, 0}]). Suppose that 1− γ > m
so that R∗ > Rs. We have that withdrawals are x = 0 for R > R∗, x = 1 − γ for R = R∗, and
x = 1 for R < R∗. Whenever R > Rs the central bank will help avoiding failure and evaluating
the collateral as if x = 0. This effectively changes the failure point to R∗ = Rs.
Central bank intervention can take the form of open market operations that reduce the fire sales
premium, or discount window lending at a very low rate.
The intervention with open market operations makes sense if a high λ is due to a temporary
spike of the market rate, that is, a liquidity crunch. In this situation a liquidity injection by the
central bank will reduce the fire sales premium. For example, after September 11 open market
operations by the Federal Reserve accepted dealers’ bids at levels well below the Federal Funds
Rate target and pushed the effective lending rate to lows of zero in several days.34
34See Markets Group of the Federal Reserve Bank of New York (2002). Martin (2002) contrasts theclassical prescription of lending at a penalty rate with the Fed’s response to September 11, namely tolend at a very low interest rate. He argues that penalty rates were needed in Bagehot’s view because theGold Standard implied limited reserves for the central bank.
27
The intervention with the discount window, perhaps more in the spirit of Bagehot, makes sense
when λ is interpreted as an adverse selection premium. The situation when a large number
of banks is in trouble displays both liquidity and adverse selection components. In any case,
the central bank intervention should be a very low rate, in contrast with Bagehot’s doctrine of
lending at a penalty rate.35 This type of intervention may provide a rationale for the apparently
strange behavior of the Federal Reserve of lending below the market rate (but with a ”stigma”
associated to it so that banks use it only when they can not find liquidity in the market).36 In
Section 7 we will provide a welfare objective for this discount window policy.
In some circumstances the central bank may not be able to infer R exactly because of noise (be
it in the supervisory process or in the observation of withdrawals). Then the central bank will
only obtain an imperfect signal of R. In this case the central bank will not be able to distin-
guish perfectly between illiquid and insolvent banks (as in Goodhart and Huang (1999a)) and,
whatever the lending policy chosen, taxpayers’ money may be involved with some probability.
This situation is realistic given the difficulty in distinguishing between solvency and liquidity
problems.37
It may be argued also that our LLR function could be performed by private banks through
credit lines . Banks providing a line of credit to another bank would then have an incentive
to monitor the borrowing institution and reduce the fire sales premium. The need for a LLR35Typically, the lending rate is kept at a penalty level to discourage arbitrage and perverse incentives.
Those considerations lie outside the present model. For example, in a repo operation the penalty for notreturning the cash on loan is to keep paying the lending rate. If this lending rate is very low the incentiveto return the loan is very small. See Fischer (1999) for a discussion of why lending should be at a penaltyrate.
36The discount window policy of the Federal Reserve is to lend 50 basis points below the target FederalFunds Rate.
37We may even think that the central bank cannot help ex post once withdrawals have materializedbut that it receives a noisy signal sCB about R at the same time as investors. The central bank then canact preventively and inject liquidity into the bank contingent on the signal received L(sCB). In this casealso the risk exists that an insolvent bank ends up being helped. The game played by the fund managerschanges, obviously, because of the liquidity injection of a large actor like the central bank.
28
remains but it may be privately provided. Goodfriend and Lacker (1999) draw a parallel between
central bank lending and private lines of credit and put emphasis on the commitment problem
of the central bank to limit lending.38 However, the central bank typically acts as LLR in
most economies probably because it has a natural superiority in terms of financial capacity
and supervisory knowledge.39 For example, in the LTCM case it may be argued that the New
York Fed had access to information that the private sector, even the members of the lifeboat
operation, did not. This unique capacity to inspect a financial institution might have made
possible the lifeboat operation orchestrated by the New York Fed. An open issue is whether this
superior knowledge continues to hold in countries where the supervision of banks is basically in
the hands of independent regulators like the Financial Services Authority (FSA) in the UK or
other countries.40
7 Endogenizing the liability structure and crisis res-
olution
In this section we endogenize the short term debt contract assumed in our model according to
which depositors can withdraw at τ = 1 or otherwise wait until τ = 2. We have seen that the
ability of investors to withdraw at τ = 1 creates a coordination problem. We argue here that
this potentially inefficient debt structure may be the only way investors can discipline a bank
manager subject to a moral hazard problem.
Suppose indeed that investment in risky assets requires the supervision of a bank manager and
that the distribution of returns of the risky assets depends on the effort undertaken by the38If this commitment problem is very acute then the private solution may be superior. However Good-
friend and Lacker (1999) do not take position on this issue. They state that “We are agnostic about theultimate role of CB lending in a welfare-maximizing steady state”.
39One of the few exceptions is the Liquidity Consortium in Germany with both participation of privatebanks and the Central Bank.
40See Vives (2001) for the workings of the FSA in the UK and its relation with the Bank of England.
29
manager. For example, the manager can either exert or not exert effort, e ∈ {0, 1} , and R
∼ N(R0, α−1) when e = 0, and R ∼ N(R, α−1) when e = 1 with R > R0. That is, exerting
effort yields a return distribution that first order stochastically dominates the one obtained by
not exerting effort. The bank manager incurs in a cost if he chooses e = 1; if he chooses e = 0 the
cost is 0. The manager also receives a benefit from continuing the project until date 2. Assume
for simplicity that the manager does not care about monetary incentives. The manager’s effort
cannot be observed so his willingness to undertake effort will depend on the relationship between
his effort and the probability that the bank continues at date 1. Withdrawals may enforce then
the early closure of the bank and provide incentives to the bank manager.41
In the banking contract, short term debt/demandable deposits can improve upon long term
debt/nondemandable deposits. With long term debt incentives cannot be provided to the man-
ager, because there is never liquidation, and therefore the manager does not exert effort. Fur-
thermore, incentives cannot be provided either with renegotiable short term debt because early
liquidation is ex post inefficient. Dispersed short term debt (i.e. uninsured deposits) is what is
needed.
Let us assume that it is worthwhile to induce the manager to exert effort. This will be true
for R − R0 large enough and the (physical) cost of asset liquidation not too large. Recall that
the (per unit) liquidation value of its assets is νR, with ν � 11+λ , whenever the bank is closed
at τ = 1. We assume, as in the previous sections, that the face value of the debt contract is
the same in periods t = 1, 2 (equal to D) and we suppose also that investors in order to trust
their money to fund managers need to be guaranteed a minimum expected return, equal to zero
without loss of generality.41This approach is based on Grossman and Hart (1982) and is followed in Gale and Vives (2002). See
also Calomiris and Kahn (1991), Diamond and Rajan (1997) and Carletti (1999).
30
The banking contract will have short-term debt and will maximize the expected profits of the
bank, choosing the investment in risky and safe assets and deposit payment, subject to the
resource constraint 1 + E = I + Dm (where Dm = M is the amount of liquid reserves held by
the bank), the incentive compatibility constraint of the bank manager, and the (early) closure
rule associated with the (unique) equilibrium in the investors’ game. This early closure rule is
defined by the property: x(R, t∗)D > M + IR1+λ , which is satisfied if and only if R < REC(t∗).
As stated before, REC(t∗) < R∗ since early closure implies failure, while the converse is not
true. Let Ro be the smallest R that fulfills the incentive compatibility constraint of the bank
manager. We have thus REC(t∗) ≥ Ro. The banking program will maximize the expected value
of the bank assets which consists of two terms: the product of the size I = 1 + E − Dm of the
bank’investments by the net expected return on these investments, taking into account expected
liquidation costs, and the value of liquid reserves Dm. Thus the optimal banking contract will
solve
Maxm
{(1 + E − Dm)(R − (1 − ν) E(R | R < REC(t∗(m))) Pr(R < REC(t∗(m))) + Dm
}
subject to
1. t∗(m) the unique equilibrium of the fund managers’ game, and
2. REC(t∗(m)) ≥ Ro.
Given that t∗(m), and therefore REC(t∗(m)), decrease with m, the optimal banking contract is
easy to characterize. If the net return on banks’ assets is always larger than the opportunity
cost of liquidity –even when the banks have no liquidity at all (that is, R − (1 − ν)E(R | R <
REC(t∗(0)) Pr(R < REC(t∗(m))) > 1), then it is clear that at the optimal point m = 0. If on
31
the contrary R − (1 − ν)E(R | R < REC(t∗(0)) Pr(R < REC(t∗(0))) < 1, there is an interior
optimum. An interesting question is how the banking contract compares with the incentive
efficient solution, which we now describe.
Given that the pooled signals of investors reveal R, we can define the incentive-efficient solution
as the choice of investment in liquid and risky assets and probability of continuation at t = 1
(as a function of R) which maximize expected surplus subject to the resource constraint and the
incentive compatibility constraint of the bank manager.42 Furthermore, given the monotonicity
of the likelihood ratio φ(R |e=0)φ(R |e=1) , the optimal region of continuation is of the cutoff form. More
specifically, the optimal cutoff will be Ro, the smallest R that fulfills the incentive compatibility
constraint of the bank manager. The cutoff Ro will be (weakly) increasing with the extent of
the moral hazard problem that bank managers face.
The incentive-efficient solution solves
Maxm
{(1 + E − Dm)(R − (1 − ν)E(R | R < Ro)) Pr(R < Ro + Dm
}
where Ro is the minimal return cutoff that incentivates the bank manager. If (R− (1− ν)E(R |
R < Ro) Pr(R < Ro) > 1 we have that mo = 0. Thus at the incentive-efficient solution it is
optimal not to hold any reserves. This should come as no surprise because we assume that there
is no cost of liquidity provision by the central bank. A more complete analysis would include
such a cost and lead to an optimal combination of the LLR policy with an ex-ante regulation of
a minimum liquidity ratio.
Since REC(t∗) must also fulfill the incentive compatibility constraint of the bank manager,
we will have that at the optimal banking contract with no LLR, REC(t∗) ≥ Ro. In fact, we42We disregard here the welfare of the bank manager and that of the funds managers.
32
will typically have a strict inequality, since there is no reason that the equilibrium threshold t∗
satisfies REC(t∗) = Ro. This means that the market solution will lead to too many early closures
of banks. This comes from the fact that the banking contract with no LLR intervention uses an
inefficient instrument (the liquidity ratio) to provide indirect incentives for bankers through the
threat of early liquidation.
The role of a modified LLR can be viewed, in this context, as correcting these market ineffi-
ciencies while maintaining the incentives of bank managers. By announcing its commitment to
provide liquidity assistance (at a zero rate) in order to avoid inefficient liquidation at τ = 1
(that is, for R > Ro) the LLR can implement the incentive efficient solution. When offered help
the bank will borrow the liquidity it needs, D(x − m)+.43
In order to implement the incentive efficient solution the modified LLR has to care about avoiding
inefficient liquidation at τ = 1 in the range (Ro, REC) and not about avoiding failure of the
bank. Now the solvency threshold Rs has no special meaning. Indeed, Ro will typically be
different from Rs. The reason is that Rs is determined by the promised payments to investors,
cash reserves and investment in the risky asset, while Ro is just the minimum threshold that
incentivates the banker to behave. We will have that Ro > Rs when the moral hazard problem
for bank managers is severe and Ro < Rs when the moral hazard problem for bank managers is
moderate.
This modified LLR facility leads to a view on the LLR that differs from Bagehot’s doctrine and43We could also envision help by the central bank in an ongoing crisis to implement the incentive
efficient closure rule. The central bank would lend then at a very low interest rate to illiquid banksfor the amount that they could not borrow in the interbank market in order to meet their paymentobligations at τ = 1. It is easy to see that in this case the equilibrium between fund managers is notmodified. This is so because central bank intervention does not change the instances of failure of thebank (indeed, when a bank is helped at τ = 1 because x(R, t∗)D > M + IR
1+λ , it will fail at τ = 2). Inthis case the coordination failure is not eliminated but its effects (on early closure) are neutralized bythe intervention of the central bank. The modified LLR helps the bank in the range (Ro, REC(t∗)) in theamount Dx(t∗, R)−(M + IR
1+λ ) > 0. LLR help (bail-out) complements the money raised in the interbankmarket IR
1+λ (bail-in).
33
introduces interesting policy questions.
Whenever Ro > Rs there is a region (specifically, for R in (Rs, Ro)) where there should be
early intervention (or prompt corrective action, to use the terminology of banking regulators).
Indeed, in this region the bank is solvent but it should be intervened to control moral hazard of
the banker. On the other hand, in the range (Ro, REC) a LLR policy is efficient if the central
bank can commit. If it cannot and instead optimizes ex post (be it because to build a reputation
is not possible or because of weakness in the presence of lobbying), it will intervene too often.
Some additional institutional arrangement is needed in the range (Rs, Ro) to implement prompt
corrective action (i.e. early closure of banks that are still solvent).
When Ro < Rs, there is a range (Ro, REC) where the bank should be helped even though
it might be insolvent (ans in this case money is lost). More precisely, for R in the range
(Ro, min{Rs, REC}) the bank is insolvent and should be helped. If the central bank’s char-
ter specifies that it cannot lend to insolvent banks then another institution (Deposit Insurance
Fund, Regulatory Agency, Treasury) financed by other means (insurance premiums or taxation)
is needed to provide an ”orderly resolution of failure” when R is in the range (Ro, min{Rs, REC}).
This could be interpreted, as in corporate bankruptcy practice, as a way to preserve the going-
concern value of the institution as well as allowing its owners and managers a fresh start after
the crisis.
An important implication of our analysis is the complementarity between bail-ins (interbank
market) and bail-outs (LLR) as well as other regulatory facilities (prompt corrective action,
orderly resolution of failure) in crisis management.
In summary we can compare different organizations:
• With neither a LLR nor an interbank market, liquidation takes place whenever x > mD,
34
which limits inefficiently investment I.
• With an interbank market but no LLR (as advocated by Goodfriend and King) the closure
threshold is REC and there is excessive failure whenever REC > Ro.
• With both a LLR facility and an interbank market:
– When Ro > Rs (severe moral hazard problem for the banker) the incentive-efficient
solution can be implemented complementing the LLR with a policy of prompt cor-
rective action in the range (Rs, Ro).
– When Ro < Rs (moderate moral hazard problem for the banker), a different insti-
tution (financed by taxation or by insurance premiums) is needed to complement
the central bank and implement the incentive-efficient solution. The central bank
helps whenever the bank is solvent and the other institution provides an ”orderly
resolution of failure” in the range (Ro, min{Rs, REC}).
8 An International Lender of Last Resort
In this section we reinterpret the model in an international setting and provide a potential
rationale for an International Lender of Last Resort (ILLR) a la Bagehot.
Financial and banking crises, usually coupled with currency crises, have been common in emerg-
ing economies in Asia (Thailand, Indonesia, Korea), Latin America (Mexico, Brazil, Ecuador,
Argentina) as well as in the periphery of Europe (Turkey). Those crisis have proved costly in
terms of output. The question is whether an ILLR can help alleviate, or avoid, those crises.
An ILLR can follow a policy of injecting liquidity in international financial markets (going from
the proposal of establishing a global central bank issuing an international currency to the mere
35
coordination of the intervention of the three major central banks44) or can act to help countries
in trouble much like a central bank acts to help individual banking institutions. The last ap-
proach is developed in several proposals that adapt Bagehot’s doctrine to international lending
(see, for example, the Meltzer Report (IFIAC (2000)) and Fischer (1999)). As pointed out by
Jeanne and Wyplosz (2001), a major difference between the approaches is on the required size of
the ILLR. In the first case an issuer of international currency is needed while in the second the
intervention is bounded by the difference between the short-term foreign exchange liabilities of
the banking sector and the foreign reserves of the country in question. We will look here at the
second approach. The main tension identified in the debate is between those who put emphasis
in the crisis prevention effect of liquidity support (Fischer (1999)) and those who are worried
about generating moral hazard in the country being helped (Meltzer Report, IFIAC (2000)).
8.1 A reinterpretation of the model
Suppose now that the balance sheet of Section 2 corresponds to a small open economy where
D0 is the foreign denominated short-term debt, M is the amount of foreign reserves, I is the
investment in risky local entrepreneurial projects, E equity and long-term debt (or local resources
available for investment), and D is the face value of the foreign denominated short-term debt.45
Our fund managers are now international fund managers operating in the international interbank
market. The liquidity ratio m = M/D is now the ratio of foreign reserves to foreign short-term
debt, a crucial ratio, according to empirical work, in determining the probability of a crisis in the
country.46 The parameter λ represents now the fire sales premium associated to early sales of44See Eichengreen (1999) for a survey of the different proposals.45The balance sheet corresponds to the consolidated private sector of the country. In some countries
local firms borrow from local banks and then the latter borrow in international currency.46Indeed, Radelet and Sachs (1998), and Rodrik and Velasco (1999) find that the ratio of short-term
debt to reserves is a robust predictor of financial crisis (in the sense of a sharp reversal of capital flows).The latter also find that a greater short-term exposure aggravates the crisis once capital flows reverse.
36
domestic bank assets in the secondary market. Furthermore, for a given amount of withdrawals
by fund managers x > m at τ = 1, there are critical thresholds for the return R of investment
below which the country is bankrupt (Rf (x)) or will default at τ = 1 (Rec(x) < Rf (x)). The
effort e necessary to improve returns could be understood to be exerted by bank managers,
entrepreneurs, or even the government. According to Section 7, effort has a cost and the actors
exerting effort are interested in continuing in their job. Default by the country at τ = 1
deprives those actors from their continuation benefits (for example, because of restructuring
of the banking and/or private sectors or because the goverment is removed from office) and
consequently ”default” at τ = 1 for some region of realized returns is the only disciplining
device.
8.2 Results
• There is a range or realizations of the return R, (Rs, R∗) , for which a coordination failure
occurs. This happens when the amount of withdrawals by foreign fund managers is so
large that the country is bankrupt even though it is (in principle) solvent.
• For a high enough foreign reserve ratio m there is no coordination failure of international
investors.
• The probability of bankruptcy of the banking sector is:
– decreasing in the foreign reserve ratio, the solvency ratio, the relative reputation
cost of withdrawal for international fund managers (C/B), and the expected mean
return of the country investment;
– increasing in the fire sales premium and the face value of foreign short-term debt;
and
37
– increasing in the precision of public information about R when public news are bad
and decreasing in the precision of private information (both provided C/B is not too
large).
• An ILLR that follows Bagehot’s prescription can minimize the incidence of coordination
failure among international fund managers provided that he is well informed about R. One
possibility is that the ILLR does in depth country research and has supervisory knowledge
of the banking system of the country where the crisis occurs.47
• The disclosure of a public signal about country return prospects may introduce multiple
equilibria. A well-informed international agency may want to be cautious and not disclose
publicly too precise information to avoid a rally of expectations in a run equilibrium.
• In the presence of a moral hazard problem to elicit high returns, foreign short-term debt
serves the purpose of disciplining whoever has to exert effort to improve returns. Note
that domestic currency denominated short-term debt will not have a disciplining effect
because it can be inflated away. There will be an optimal cutoff point Ro below which
restructuring must happen (be it of the private sector or government) in order to provide
incentives to exert effort.
• The following scenarios can be considered:
– No bail-in, no bail-out. With no ILLR and no access to the international inter-
bank market, country projects are liquidated whenever withdrawals by foreign fund
managers are larger than foreign reserves. This limits inefficiently investment.
– Bail-in but no bail-out. With no ILLR but access to the international interbank47Although this seems more farfetched than in the case of a domestic LLR, the IMF, for example, is
trying to enhance its monitoring capabilities with the Financial Sector Assessment Programs.
38
market, some costly project liquidation is avoided with fire sales of assets but still
there will be excessive liquidation of entrepreneurial projects.
– Bail-in and bail-out. With ILLR and access to the international interbank market:
∗ When the moral hazard problem in the country is severe (Ro > Rs), a policy
of prompt corrective action in the range (Rs, Ro) is needed to complement the
ILLR facility. A solvent country may need to ”restructure” when returns are
close to the solvency threshold.
∗ When the moral hazard problem in the country is moderate (Ro < Rs), on top
of the ILLR help for a solvent country, an orderly resolution of failure process is
needed in the range (Ro, min{Rs, REC}). An insolvent country should be helped
when not too far away from the solvency threshold. This may be interpreted
as a mechanism similar to the sovereign debt restructuring mechanim (SDRM)
of the sort currently studied by the IMF with the objective of restructuring
unsustainable debt.48 In our case this would be the foreign short-term debt. In
the range (Ro, min{Rs, REC}) an institution like an international bankruptcy
court could help.
As before an important insight from the analysis is the complementarity between the market
(bail-ins) and an ILLR facility (bail-out) together with other regulatory facilities to provide
for prompt corrective action and orderly failure resolution. Our conclusion is that an ILLR
facility a la Bagehot can help implementing the incentive efficient solution provided that it is
complemented with prompt corrective action and orderly resolution of failure provisions.48See Bolton (2002) for a discussion of SDRM type facilities from the perspective of corporate
bankruptcy theory and practice.
39
9 Concluding remarks
In this paper we have provided a rationale for Bagehot’s doctrine of helping illiquid but solvent
banks in the context of modern interbank markets. Indeed, investors in the interbank market
may face a coordination failure and intervention may be desirable. We have examined the impact
of public intervention along the following three dimensions:
• solvency and liquidity requirements (at τ = 0);
• Lender of Last Resort policy (at the interim date τ = 1); and
• closure rules, which can consist of two types of policy: orderly resolution of failures or
prompt corrective action.
The coordination failure can be avoided by appropriate solvency and liquidity requirements.
However, the cost of doing so will typically be too large in terms of foregone returns and ex
ante measures will only help partially. This means that prudential regulation needs to be
complemented by a Lender of Last Resort policy. The paper shows how discount window loans
can eliminate the coordination failure (or alleviate it, if for incentive reasons some degree of
coordination failure is optimal). It also sheds light on when open market operations will be
appropriate.
A main insight of the analysis is that public and private involvement are complementary in im-
plementing the incentive efficient solution. Furthermore, the implementation of this solution may
require also to complement Bagehot’s LLR facility with prompt corrective action (intervention
on a solvent bank) or orderly failure resolution (help to an insolvent bank).
The model, when given an interpretation in an international context, provides a rationale for
an international LLR a la Bagehot, complemented with prompt corrective action and orderly
40
failure resolution provisions, and points at the complementarity between bail-ins and bail-outs
in crisis resolution.
41
References
Allen, Franklin and Douglas Gale (1998), “Optimal Financial Crises”, Journal of
Finance, 53, pp. 1245-1284.
Bagehot, Walter (1873), Lombard Street, London, H.S. King.
Bernanke, Ben (1983), ”Nonmonetary Effects of the Financial Crisis in the Propa-
gation of the Great Depression”, American Economic Review, 73, pp. 257-263.
Bernanke, Ben and Mark Gertler (1989), ”Agency Costs, Net Worth, and Business
Fluctuations,” American Economic Review, 79, pp. 14-31.
Berger, Allen, Sally Davies and Mark Flannery (1998), ”Comparing Market and
Supervisory Assessments of Bank Performances: Who Knows What When?”,
Finance and Economics Discussion Series no. 1998-32, Federal Reserve Board
of Governors.
Bolton, Patrick (2002), ”Towards a Statutory Approach to Sovereign Debt Re-
structuring: Lessons from Corporate Bankruptcy Practice around the World”,
mimeo, Princeton University.
Bordo, Michael D. (1990), “The Lender of Last Resort: Alternative Views and
Historical Experience”, Federal Reserve Bank of Richmond, Economic Review,
pp. 18-29.
Broecker, Thoersten (1990), ”Credit-Worthiness Tests and Interbank Competition”,
Econometrica, 58, 2, pp. 429-452.
Bryant, James (1980), “A Model of Reserves, Bank Runs and Deposit Insurance”,
Journal of Banking and Finance, 4, pp. 335-344.
42
Calomiris, Charles (1988a), “The IMF’s Imprudent Role as a Lender of Last Resort”,
Cato Journal, 17, 2, February.
Calomiris, Charles (1988b), “Blueprints for a New Global Financial Architecture”,
Joint Economics Committee, United States Congress, Washington D.C. (Octo-
ber 7th).
Calomiris, Charles and Charles Kahn (1991), “The Role of Demandable Debt in
Structuring Optimal Banking Arrangements”, American Economic Review ,
81(3), pp. 497-513.
Carletti, Elena (1999), ”Bank Moral Hazard and Market Discipline”, FMG Discus-
sion Paper, LSE, London UK.
Carlsson, Hans and Eric Van Damme (1993), “Global Games and Equilibrium Se-
lection”, Econometrica, 61(5), pp. 989-1018.
Chari, Varadarajan V. and Ravi Jagannathan (1988), “Banking Panics, Informa-
tion, and Rational Expectations Equilibrium”, Journal of Finance, 43(3), pp.
749-761.
Chari, Varadarajan V. and Patrick Kehoe (1998), ”Asking the Right Questions
about the IMF,” Public Affairs, 13, pp. 3-26.
Corsetti, Giancarlo, Amil Dasgupta, Stephen Morris and Hyun S. Shin (2000),
”Does One Soros make a Difference? The Role of a Large Trader in Currency
Crises”, mimeo, LSE, London UK.
DeYoung, Robert, Mark Flannery, Walter Land and Sorin Sorescu (1998), ”The In-
formational Advantage of Specialized Monitors: The Case of Bank Examiners”,
mimeo.
43
Diamond, Douglas and Phil Dybvig (1983), “Bank Runs, Deposit Insurance and
Liquidity”, Journal of Political Economy, 91, 3, pp. 401-419.
Diamond, Douglas and Ragu Rajan (2001), ”Liquidity Risk, Liquidity Creation and
Financial Fragility : A Theory of Banking”, Journal of Political Economy, 109,
2, pp. 287-327.
Eichengreen, Barry, Toward a New International Financial Architecture: A Practi-
cal Post-Asia Agenda, Institute for International Economics, 1999.
Fischer, Stanley (1999), ”On the Need for an International Lender of Last Resort,”
The Journal of Economic Perspectives, 13, pp. 85-104.
Flannery, Mark (1996), ”Financial Crises, Payment Systems Problems, and Dis-
count Window Lending”, Journal of Money, Credit and Banking 28, 4, pp.
804-824.
Folkerts-Landau, Dieter and Peter Garber (1992), ”The ECB: a Bank or a Monetary
Policy Rule?”, in M.B. Canzoneri, V. Grilland, P.R. Masson, (Eds), CEPR,
Cambridge University Press, Establishing a Central Bank: Issues in Europe and
Lessons from the US, Chapter 4, pp. 86-110.
Freixas, Xavier, Curzio Giannini, Glenn Hoggarth and Farouk Soussa (1999) ”Lender
of Last Resort-A Review of the Literature”Financial Stability Review, Bank of
England, 7, pp. 151-167.
Freixas, Xavier, Bruno Parigi and Jean-Charles Rochet (2003), ”The Lender of Last
Resort : A 21st Century Approach”, mimeo, University of Toulouse, France.
Gale, Douglas and Xavier Vives (2002), “Dollarization, Bailouts, and the Stability
of the Banking System”, Quarterly Journal of Economics, 117, 2, pp. 467-502.
44
Goldstein, Itay and Ari Pauzner (2000), ”Demand Deposit Contracts and the Prob-
ability of Bank Runs”, Discussion Paper, Tel-Aviv University.
Goodfriend, Marvin and Robert King (1988), “Financial Deregulation, Monetary
Policy and Central Banking”, in Restructuring Banking and Financial Services
in America, Haraf W. and R.M. Kushmeider eds., AEI Studies , 481, Lanham
Md, USA.
Goodfriend, Marvin and Jeff Lacker (1999), ”Limited Commitment and Central
Bank Lending”, Federal Reserve Bank of Richmond WP 99-2.
Goodhart, Charles (1995), The Central Bank and the Financial System, Cambridge,
MA : MIT Press.
Goodhart, Charles and Haizhou Huang (1999a), “A Model of the Lender of Last
Resort”, FMG, LSE Discussion Paper 313, London, U.K..
Goodhart, Charles and Haizhou Huang (1999b), ”A Simple Model of an Interna-
tional Lender of Last Resort”, FMG, LSE Discussion Paper 336, London, U.K..
Gorton, Gary (1985), “Bank Suspension of Convertibility”, Journal of Monetary
Economics, pp. 177-193.
Gorton, Gary (1988), ”Banking Panics and Business Cycles”, Oxford Economic
Papers 40, pp. 751-781.
Grossman, Sandy and Oliver Hart (1982), ”Corporate Financial Structure and Man-
agerial Incentives,” in J. McCall (ed.) The Economics of Information and Un-
certainty, pp. 107-137. Chicago: University of Chicago Press.
Hart, Oliver and John Moore (1994), ”A Theory of Debt Based on the Inalienability
of Human Capital”, Quarterly Journal of Economics, 109, 4, pp. 841-879.
45
Heinemann, Frank and Gerhard Illing (2000), ”Speculative Attacks: Unique Sunspot
Equilibrium and Transparency”, mimeo, CFS, Frankfurt, Germany.
Hellwig, Christian (2002),”Public Information, Private Information and teh Multi-
plicity of Equilibria in Coordination Games”, Journal of Economic Theory, 107,
pp. 191-222.
Holmstrom, Bengt and Jean Tirole (1997), ”Financial Intermediation, Loanable
Funds, and the Real Sector”, Quarterly Journal of Economics, 112, 3, pp. 663-
691.
Humphrey, Thomas (1975), “The Classical Concept of the Lender of Last Resort”,
Economic Review, 61.
International Financial Institution Advisory Commission (2000), Report of the In-
ternational Financial Institution Advisory Commission, Allan H. Meltzer, Chair-
man, Washington DC, March.
Jacklin, Charles and Sudipto Bhattacharya (1988), “Distinguishing Panics and
Information-Based Bank Runs: Welfare and Policy Implications”, Journal of
Political Economy 96(3), pp. 568-592.
Jeanne, Oliver and Charles Wyplosz (2001), ”The International Lender of Last
Resort: How Large is Large Enough?”, NBER WP 8381.
Kaminsky, Graciela L. and Carmen Reinhart, (1999), ”The Twin Crises: The Causes
of Banking and Balance-of-Payments Problems”, American Economic Review ,
89, pp. 473-500.
Kaufman, George (1991), “Lender of Last Resort: a Contemporary Perspective”,
Journal of Financial Services Research, 5, pp. 95-110.
46
Lindgren, Carl-Johan, Gillian Garcia and Matthew Saal (1996), Bank Soundness
and Macroeconomic Policy, IMF, Washington, D.C.
Markets Group of the Federal Reserve Bank of New York (2002), ”Domestic Open
Market Operations During 2001”, Federal Reserve Bank of New York.
Martin, Antoine (2002) ” Reconciling Bagehot with the Fed’s response to Sept. 11”,
FRB of Kansas City, October.
Mishkin, Frederick (1998), “Systemic Risk, Moral Hazard and the International
Lender of Last Resort”, NBER and Columbia University, April.
Morris, Stephen and Hyun S. Shin (1998), “Unique Equilibrium in a Model of Self-
Fulfilling Currency Attacks”, American Economic Review, 88, pp. 587-597.
Morris, Stephen and Hyun S. Shin (2000), ”Rethinking Multiple Equilibria in Macroe-
conomic Modelling”, NBER Macroeconomics Annual.
Morris, Stephen and Hyun S. Shin(2001), ”The CNBC Effect: Welfare Effects of
Public Information”, mimeo.
Padoa-Schioppa, Tomaso (1999), ”EMU and Banking Supervision”, Lecture at the
London School of Economics, Financial Market Group.
Peek, Joe, Eric Rosengren and Geoffrey Tootell (1999), ”Is Bank Supervision Central
to Central Banking”, Quarterly Journal of Economics, 114 (2), pp. 629-53.
Postlewaite, Andy and Xavier Vives (1987), “Bank Runs as an Equilibrium Phe-
nomenon”, Journal of Political Economy, 95(3), pp. 485-491.
Radelet, Steven and Jeffrey Sachs (1998), ”The Onset of the Asian Financial Crisis”,
mimeo, Harvard Institute for International Development.
Riordan, Mike (1993), ”Competition and Bank Performance: A Theoretical Per-
47
spective” in Capital Markets and Financial Intermediation (C. Mayer and X.
Vives eds.), pp. 328-348, Cambridge: Cambridge University Press.
Rochet,Jean-Charles and Xavier Vives (2002), “Coordination Failures and the Lender
of Last Resort: Was Bagehot Right After All?”, CEPR DP 3233.
Rodrik, Dani and Andres Velasco (1999), ”Short-Term Capital Flows”, World Bank
1999 ABCDE Conference.
Romer, Christina and David Romer (2000), ”Federal Reserve Information and the
Behavior of Interest Rates”, American Economic Review, 90, 3, pp. 429-453.
Schwartz, Anna J. (1992), ”The Misuse of the Fed’s Discount Window”, Federal
Reserve Bank of St. Louis Review, September/October, pp. 58-69.
Tarkka, Juha and David Mayes (2000), ”The Value of Publishing Official Central
Bank Forecasts”, mimeo, Bank of Finland.
Thornton, Henry (1802), An Enquiry into the Nature and Effects of Paper Credit
of Great Britain, London, Hatchard.
Van Zandt, Timothy and Xavier Vives (2003), “Monotone Equilibria in Bayesian
Games of Strategic Complementarities”, INSEAD WP.
Vives, Xavier (1990),”Nash Equilibrium with Strategic Complementarities”, Jour-
nal of Mathematical Economics, 19, 3, pp. 305-321.
Vives, Xavier (1999), Oligopoly Pricing: Old Ideas and New Tools, MIT Press,
Boston and London.
Vives, Xavier (2001), “Restructuring Financial Regulation in the European Mone-
tary Union”, Journal of Financial Services Research, 19, 1, pp. 57-82.
48